Estimation and Inference for Set-identi ed Parameters Using Posterior Lower Probability

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1 Estimation and Inference for Set-identi ed Parameters Using Posterior Lower Probability Toru Kitagawa Department of Economics, University College London 28, July, 2012 Abstract In inference for set-identi ed parameters, Bayesian probability statements about unknown parameters do not coincide, even asymptotically, with frequentist s con dence statements. This paper aims to smooth out this disagreement from a robust Bayes perspective. I show that a class of prior distributions exists, with which the posterior inference statements drawn via the lower envelope (lower probability) of the class of posterior distributions asymptotically agrees with frequentist con dence statements for the identi ed set. With this class of priors, the statistical decision problems, including the point and set estimation of the set-identi ed parameters, are analyzed under the posterior gamma-minimax criterion. Keywords: Partial Identi cation, Bayesian Robustness, Belief Function, Imprecise Probability, Gamma-minimax, Random Set. t.kitagawa@ucl.ac.uk. I thank Gary Chamberlain, Andrew Chesher, Siddhartha Chib, Larry Epstein, Jean-Pierre Florens, Guido Imbens, Hiro Kaido, Charles Manski, Ulrich Müller, Andriy Norets, Adam Rosen, Kevin Song, and Elie Tamer for their valuable discussions and comments. I also thank the seminar participants at Academia Sinica, Brown, Cornell, Cowles Conference 2011, EC , Harvard/MIT, Midwest Econometrics 2010, Northwestern, NYU, RES Conference 2011, Seoul National University, Simon Fraser University, and UBC for their helpful comments. All remaining errors are mine. Financial support from the ESRC through the ESRC Centre for Microdata Methods and Practice (CeMMAP) (grant number RES ) is gratefully acknowledged. 1

2 1 Introduction In inferring identi ed parameters in a parametric setup, the Bayesian probability statements about unknown parameters are found to be similar, at least asymptotically, to the frequentist con dence statements about the true value of the parameters. In partial identi cation analyses initiated by Manski (1989, 1990, 2003, 2007), such asymptotic harmony between the two inference paradigms breaks down (Moon and Schorfheide (2011)). The Bayesian interval estimates for the set-identi ed parameter are shorter, even asymptotically, than the frequentist ones, and they asymptotically lie inside the frequentist con dence intervals. Frequentists might interpret this phenomenon, Bayesian over-con dence in their inferential statements, as being ctitious. Bayesians, on the other hand, might consider that the frequentist con dence statements, which apparently lack posterior probability interpretation, raise some interpretative di culty once data are observed. The primary aim of this paper is to smooth out the disagreement between the two schools of statistical inference by applying the perspective of a robust Bayes inference, where one can incorporate partial prior knowledge into posterior inference. While there is a variety of robust Bayes approaches, this paper focuses on a multiple prior Bayes analysis, where the partial prior knowledge, or the robustness concern against prior misspeci cation, is modeled with a class of priors (ambiguous belief). The Bayes rule is applied to each prior to form a class of posteriors. The posterior inference procedures considered in this paper operate on the class of posteriors by focusing on their lower and upper envelopes, the so-called posterior lower and upper probabilities. When the parameters are not identi ed, the prior distribution of the model parameters can be decomposed into two components: one that can be updated by data (revisable prior knowledge) and one that can never be updated by data (unrevisable prior knowledge). Given that the ultimate goal of the partially identi cation analysis is to establish a "domain of consensus" (Manski (2007)) among the set of assumptions that data are silent about, a natural way to incorporate this agenda into the robust Bayes framework is to design a prior class in such a way that it shares a single prior distribution for the revisable prior knowledge, but allows for arbitrary prior distributions for the unrevisable prior knowledge. Using this prior class as a prior input, this paper derives the posterior lower probability and investigates 2

3 its analytical property. For an interval-identi ed parameter case, I also examine whether the inferential statements drawn via the posterior lower probability can asymptotically have any type of valid frequentist coverage probability in the partially identi ed setting. Another question this paper examines is, with such class of priors, how to formulate and solve statistical decision problems including point estimation of the set-identi ed parameters. I approach this question by adapting the posterior gamma-minimax analysis, which can be seen as a minimax analysis with the multiple posteriors, and demonstrate that the proposed prior class leads to an analytically tractable and numerically solvable formulation of the posterior gamma-minimax decision problem, provided that the identi ed set for the parameter of interest can be computed for each possible distribution of data. 1.1 Related Literature Estimation and inference in partially identi ed models are a growing research area in the eld of econometrics. From the frequentist perspective, Horowitz and Manski (2000) construct con dence intervals for an interval identi ed set. Imbens and Manski (2004) propose uniformly asymptotically valid con dence sets for an interval-identi ed parameter, which are further extended by Stoye (2009). Chernozhukov, Hong, and Tamer (2007) develop a way to construct asymptotically valid con dence sets for an identi ed set based on the criterion function approach, which can be applied to a wide range of partially identi ed models including moment inequality models. In relation to the criterion function approach, the literature on the construction of con dence sets by inverting test statistics includes, but is not limited to, Andrews and Guggenberger (2009), Andrews and Soares (2010), and Romano and Shaikh (2010). From the Bayesian perspective, Neath and Samaniego (1997), Poirier (1998), and Gustafson (2009, 2010) analyze how Bayesian updating performs when a model lacks identi cation. Liao and Jiang (2010) conduct a Bayesian inference for moment inequality models, based on the pseudo-likelihood. Moon and Schorfheide (2011) compare the asymptotic properties of frequentist and Bayesian inferences for set-identi ed models. My robust Bayes analysis is motivated by Moon and Schorfheide s important ndings on the asymptotic disagreement between the frequentist and Bayesian inferences. Epstein and Seo (2012) focus on a set- 3

4 identi ed model of entry games with multiple equilibria, and provide an axiomatic argument that justi es a single-prior Bayesian inference for a set-identi ed parameter. The current paper does not intend to provide any normative argument as to whether one should proceed with a single prior or multiple priors in inferring non-identi ed parameters. The analysis of lower and upper probabilities originates with Dempster (1966, 1967a, 1967b, 1968), in his ducial argument of drawing posterior inferences without specifying a prior distribution. The in uence of Dempster s appears in the belief function analysis of Shafer (1976, 1982) and the imprecise probability analysis of Walley (1991). In the context of robust Bayes analysis, the lower and upper probabilities have been playing important roles in measuring the global sensitivity of the posterior (Berger (1984), Berger and Berliner (1986)) and also in characterizing a class of priors/posteriors (DeRobertis and Hartigan (1981), Wasserman (1989, 1990), and Wasserman and Kadane (1990)). In econometrics, pioneering work using multiple priors was carried out by Chamberlain and Leamer (1976), and Leamer (1982), who obtained the bounds for the posterior mean of the regression coe cients when a prior varies over a certain class. All of these previous studies did not explicitly consider non-identi ed models. This paper, in contrast, focuses on non-identi ed models, and aims to clarify a link between the early idea of the lower and upper probabilities and a recent issue on inferences in set-identi ed models. The posterior lower probability to be obtained in this paper is an in nite-order monotone capacity, or equivalently, a containment functional in the random set theory. Beresteanu and Molinari (2008) and Beresteanu, Molchanov, and Molinari (2012) show the usefulness and wide applicability of the random set theory to a class of partially identi ed models by viewing observations as random sets, and the estimand (identi ed set) as its Aumann expectation. They propose an asymptotically valid frequentist inference procedure for the identi ed set by employing the central limit theorem applicable to the properly de ned sum of random sets. Galichon and Henry (2006, 2009) and Beresteanu, Molchanov, and Molinari (2011) propose a use of in nite-order capacity in de ning and inferring the identi ed set in the structural econometric model with multiple equilibria. The robust Bayes analysis of this paper closely relates to the literature of non-additive measures and random sets, but the way that these theories enter to the analysis di ers from these previous works in the following ways. First, the class of models to be considered is assumed to have well- 4

5 de ned likelihood functions, and the lack of identi cation is modeled in terms of the "dataindependent at regions" of the likelihood. Ambiguity is not explicitly modeled at the level of observations, but instead ambiguity for the parameters is introduced through the absence of prior knowledge on each at region of the likelihood. Second, I obtain the identi ed set as random sets, whose probability law is represented by the posterior lower probability. Here, the source of probability that induces the random identi ed set is the posterior uncertainty for the identi able parameters, not the sampling probability of the observations. Third, the inferential statements to be proposed in the paper are made conditional on data, and they do not invoke any large-sample approximations. The decision theoretic analysis in this paper employs the posterior gamma-minimax criterion, which leads to a decision that minimizes the worst case posterior risk over the class of posteriors. The gamma-minimax decision analysis often becomes challenging, both analytically and numerically, and the existing analyses are limited to rather simple parametric models with a certain choice of prior class (Betro and Ruggeri (1992), Chamberlain (2000), and Vidakovic (2000)). The speci ed prior class, in contrast, o ers a general and feasible way to solve the posterior gamma-minimax decision problem, provided that the identi ed set for the parameter of interest can be computed for each of the identi ed parameter values. In a recent study by Song (2012), point estimation for an interval-identi ed parameter from the local asymptotic minimax approach is considered. 1.2 Plan of the Paper The rest of the paper is organized as follows. In Section 2, the main results of this paper are presented using a simple example of missing data. Section 3 introduces the general framework, where I construct a class of prior distributions that can contain arbitrary unrevisable prior knowledge. I then derive the posterior lower and upper probabilities. Statistical decision analyses with multiple priors are examined in Section 4. In Section 5, how to construct the posterior credible regions based on the posterior lower probability is discussed and their large-sample behaviors are examined in an interval-identi ed parameter case. Proofs and lemmas are provided in Appendix A. 5

6 2 An Illustration: A Missing Data Example This section illustrates the main results of the paper using an example of missing data (Manski (1989)). Let Y 2 f1; 0g be the binary outcome of interest (e.g., a worker is employed or not). Let W 2 f1; 0g be an indicator of whether Y is observed (W = 1) or not (W = 0), (i.e., the subject responded or not). Data are given by a random sample of size n, x = f(y i W i ; W i ) : i = 1; : : : ; ng. The starting point of the analysis is to specify a parameter vector 2 that pins down the distribution of the data and the parameter of interest. Here, can be speci ed by a vector of four probability masses: ( yd ), yd Pr (Y = y; D = d), y = 1; 0, and d = 1; 0. The observed data likelihood for is written as p(xj) = n n [ ] n mis ; where n 11 = P n i=1 Y iw i ; n 01 = P n i=1 (1 Y i)w i ; n mis = P n i=1 (1 W i). This likelihood function depends on only through the three probability masses, = ( 11 ; 01 ; mis ) ( 11 ; 01 ; ); 2, no matter what the observations are, so that the likelihood for has "data-independent" at regions, which are expressed as a set-valued map of, () f 2 : 11 = 11 ; 01 = 01 ; = mis g : The parameter of interest is the mean of Y; which is written as a function of, Pr(Y = 1) = The identi ed set of, H (), as the set-valued map of is de ned by the range of when varies over (), H() = [ 11 ; 11 + mis ]; which are the Manski (1989) s bounds of Pr(Y = 1). The standard Bayes inference for would proceed as follows; specify a prior of, update it using the Bayes rule, and marginalize the posterior of to. If the likelihood for has dataindependent at regions as represented by f () : 2 g, then the prior for conditional on f 2 ()g (i.e., belief on how the proportion of missing observations, mis, is divided into 10 and 00 ) will never be updated by data. Consequently, the posterior of and possibly that of become sensitive to the speci cation of such conditional priors of given. The 6

7 robust Bayes procedure considered in this paper aims to make the posterior inference free from such sensitivity concerns by introducing multiple priors for. The way to construct a prior class is as follows. I rst specify a single prior for the identi ed parameters. In view of, prior speci es how much prior belief should be assigned to each at region of the s likelihood (), whereas, depending on ways to allocate the assigned belief over 2 () (for each ), the implied prior for may di er. Therefore, by collecting all the possible ways of allocate the assigned belief over f 2 ()g for each, I can construct the following class of prior distributions of, M = : ( (B)) = (B) for all B, where denotes a prior distribution for. By applying the Bayes rule to each 2 M and marginalizing each posterior of for, I obtain the class of posteriors of, F jx : 2 M. I now summarize the class of posteriors of by its lower envelope (lower probability), F jx (D) = inf 2M( ) F jx (D), which maps subset D in the parameter space of to [0; 1]. In words, the posterior lower probability evaluated at D says that the posterior belief allocated for f 2 Dg is at least F jx (D), no matter which 2 M is used. The main theorem of this paper shows that the posterior lower probability satis es F jx (D) = F jx (f : H () Dg), where F jx denotes the posterior distribution of implied from the prior. The key insight of this equality is that, with prior class M, drawing inference for based on its posterior lower probability is done by analyzing the probability law of random sets H (), F jx. Leaving their formal analysis to the later sections of this paper, I now outline the implementation of the posterior lower probability inference for proposed in this paper. 1. Specify a prior for and update it by the Bayes rule. When a credible prior for is not available, a reasonably "non-informative" prior may be used as far as the posterior of is proper Let f s : s = 1; : : : ; Sg be random draws of from the posterior of. The mean and median of the posterior lower probability of can be de ned via the gamma-minimax 1 See Kass and Wasserman (1996) for a survey of reasonably non-informative priors. 7

8 decision criterion, and they can be approximated by 1 arg min a S respectively. SX sup (a ) 2 1 and arg min 2H( s ) a S s=1 SX sup ja j ; 2H( s ) 3. The posterior lower credible region of at credibility level 1, which can be interpreted as a (1 )- level set of the posterior lower probability of, is de ned by the smallest interval that contains H () with posterior probability 1 (Proposition 5.1 in this paper proposes an algorithm to compute the posterior lower credible region for interval-identi ed cases). s=1 Under certain regularity conditions that are satis ed in the current missing data example, the posterior lower credible region of asymptotically attains the frequentist coverage probability 1 for the true identi ed set H ( 0 ). where 0 is the value of corresponding to the sampling distribution of data. 3 Multiple-prior Analysis and the Lower and Upper Probabilities 3.1 Likelihood and Set Identi cation: The General Framework Let (X; X ) and (; A) be measurable spaces of a sample X 2 X and a parameter vector 2, respectively. The analytical framework of this paper covers both a parametric model = R d, d < 1, and a non-parametric model where is a separable Banach space. The sample size is implicit in the notation. Let be a marginal probability distribution on the parameter space (; A), referred to as a prior distribution for. Assume that the conditional distribution of X given exists and has the probability density p(xj) at every 2 with respect to a - nite measure on (X; X ). The parameter vector may consist of parameters that determine the behaviors of the economic agents, as well as those that characterize the distribution of the unobserved heterogeneities in the population. In the context of the missing data or counterfactual causal models, indexes the distribution of the underlying population outcomes or the potential outcomes. In all of these cases, the parameter should be distinguished from the parameters 8

9 that are solely used to index the sampling distribution of observations. The identi cation problem of typically arises in this context. If multiple values of generate the same distribution of data, then these s are observationally equivalent and the identi cation of fails. In terms of the likelihood function p(xj), the observational equivalence of and 0 6= means that the values of the likelihood at and 0 are equal for every possible sample, i.e., p(xj) = p(xj 0 ) for every x 2 X (Rothenberg (1971), Drèze (1974), and Kadane (1974)). I represent the observational equivalence relation of s by a many-to-one function g : (; A)! (; B): g() = g( 0 ) if and only if p(xj) = p(xj 0 ) for all x 2 X: The equivalence relationship partitions the parameter space into equivalent classes, in each of which the likelihood of is at, irrespective of observations, and = g() maps each of these equivalent classes to a point in another parameter space. In the language of structural models in econometrics (Hurwicz (1950), and Koopman and Reiersol (1950)), = g() is interpreted as the reduced-form parameter that carries all the information for the structural parameters through the value of the likelihood function. In the literature of Bayesian statistics, = g() is referred to as the minimally su cient parameter (su cient parameter for short), and the range space of g(), (; B), is called the su cient parameter space (Barankin (1960), Dawid (1979), Florens and Mouchart (1977), Picci (1977), and Florens, Mouchart, and Rolin (1990)). 2 In the presence of su cient parameters, the likelihood depends on, only through the function g(), i.e., there exists a B-measurable function ^p(xj) such that p(xj) = ^p(xjg()) 8x 2 X and 2 (3.1) holds (Lemma of Lehmann and Romano (2005)). Denote the inverse image of g() by : () = f 2 : g() = g, 2 Florens and Simoni (2011) provide comprehensive discussions on the relationship between frequentist and Bayesian identi cation. 9

10 where () and ( 0 ) for 6= 0 are disjoint, and f () ; 2 g constitutes a partition of. I assume g() = ; so () is non-empty for every 2. 3 In the set-identi ed model, the parameter of interest 2 H is a subvector or a transformation of denoted by = h(), h : (; A)! (H; D). The formal de nition of the identi ed set of is given as follows. De nition 3.1 (Identi ed Set of ) (i) The identi ed set of is a set-valued map H : H de ned by the projection of onto H through h(), H() fh() : 2 ()g : () (ii) The parameter = h() is point-identi ed at if H() is a singleton, and is setidenti ed at if H () is not a singleton. Note that the identi cation of is de ned in the pre-posterior sense because it is based on the likelihood evaluated at every possible realization of a sample, not only for the observed one. 3.2 Examples I now provide some examples, in addition to the illustrating example of Section 2, both to illustrate the above concepts and notations, and to provide a concrete focus for the later development. Example 3.1 (Bounding ATE by Linear Programming) Consider the treatment effect model with incompliance and a binary instrument 2 f1; 0g, as considered in Imbens and Angrist (1994), and Angrist, Imbens, and Rubin (1996). Assume that the treatment status and the outcome of interest are both binary. Let (W 1 ; W 0 ) 2 f1; 0g 2 be the potential treatment status in response to the instrument, and W = W 1 + (1 )W 0 be the observed treatment status. (Y 1 ; Y 0 ) 2 f1; 0g 2 is a pair of treated and control outcomes and 3 In an observationally restrictive model, in the sense of Koopman and Reiersol (1950), ^p(xj) likelihood function for the su cient parameters, is well de ned for a domain larger than g() (see Example 3.1 in Section 3.2). In this case, the model possesses the falisi ability property, and () can be empty for some 2. 10

11 Y = W Y 1 + (1 W )Y 0 is the observed outcome. Data is a random sample of (Y i ; W i ; i ). Following Imbens and Angrist (1994), consider partitioning the population into four subpopulations de ned in terms of the potential treatment-selection responses: 8 c if W 1i = 1 and W 0i = 0 : complier, >< at if W 1i = W 0i = 1 : always-taker, T i = nt if W 1i = W 0i = 0 : never-taker, >: d if W 1i = 0 and W 0i = 1 : de er, where T i is the indicator for the types of selection responses. Assume a randomized instrument,? (Y 1 ; Y 0 ; W 1 ; W 0 ). Then, the distribution of observables and the distribution of potential outcomes satisfy the following equalities for y 2 f1; 0g: Pr(Y = y; W = 1j = 1) = Pr(Y 1 = y; T = c) + Pr(Y 1 = y; T = at); (3.2) Pr(Y = y; W = 1j = 0) = Pr(Y 1 = y; T = d) + Pr(Y 1 = y; T = at); Pr(Y = y; W = 0j = 1) = Pr(Y 0 = y; T = d) + Pr(Y 1 = y; T = nt); Pr(Y = y; W = 0j = 0) = Pr(Y 0 = y; T = c) + Pr(Y 1 = y; T = nt): Ignoring the marginal distribution of, a full parameter vector of the model can be speci ed by a joint distribution of (Y 1 ; Y 0 ; T ): = (Pr(Y 1 = y; Y 0 = y 0 ; T = t) : y = 1; 0; y 0 = 1; 0; t = c; nt; at; d) 2 ; where is the 16-dimensional probability simplex. Let ATE be the parameter of interest. E(Y 1 Y 0 ) = X [Pr(Y 1 = 1; T = t) Pr(Y 0 = 1; T = t)] t=c;nt;at;d X X = [Pr(Y 1 = 1; Y 0 = y; T = t) t=c;nt;at;d y=1;0 Pr(Y 1 = y; Y 0 = 1; T = t)] h(): The likelihood conditional on depends on only through the distribution of (Y; W ) given, so the su cient parameter vector consists of eight probability masses: = (Pr(Y = y; W = wj = z) : y = 1; 0; d = 1; 0; z = 1; 0) : 11

12 The set of equations (3.2) de nes () the set of observationally equivalent distributions of (Y 1 ; Y 0 ; T ) ; when the data distribution is given at. Balke and Pearl (1997) derive the identi ed set of ATE, H() = h( ()); by maximizing or minimizing h(), subject to 2 and constraints (3.2). obtained as a convex interval. This optimization can be solved by linear programming and H() is Note that, in this model, special attention is needed for the su cient parameter space to ensure that () is non-empty. Pearl (1995) shows that the distribution of data is compatible with the instrument exogeneity condition,? (Y 1 ; Y 0 ; W 1 ; W 0 ) ; if and only if max w X y max fpr(y = y; W = w)j = zg 1: (3.3) z This implies that in order to guarantee one for that ful ll (3.3). () 6= ;, a prior distribution for puts probability Example 3.2 (Linear Moment Inequality Model) Consider the model where the parameter of interest 2 H is characterized by linear moment inequalities, E(m(X) A) 0; where the parameter space H is a subset of R L, m(x) is a J-dimensional vector of known functions of an observation, and A is a J L known constant matrix. By augmenting the J-dimensional parameter 2 [0; 1) J, these moment inequalities can be written as the J-moment equalities, 4 E(m(X) A ) = 0: To obtain a likelihood function for the current moment equality model, specify the full parameter vector to be = (; ) 2 H [0; 1) J, and consider the exponentially tilted empirical likelihood for as considered in Schennach (2005). Let x = (x 1 ; : : : ; x n ) be a size n random sample of observations, and de ne g() = A +. If the convex hull of [ i fm(x i ) g()g contains the origin, then the exponentially tilted empirical likelihood is written as p(xj) = Y w i (); 4 I owe the Bayesian formulation of the moment inequality model shown here to Tony Lancaster (personal communication 2006). 12

13 where w i () = exp f(g()) 0 (m(x i ) g())g P n i=1 exp f(g())0 (m(x i ) g())g ; (g()) = arg min 2R J + ( nx exp f 0 (m(x i ) i=1 g())g Thus, the parameter = (; ) enters the likelihood only through g() = A +. Consequently, I take = g() to be the su cient parameters. by () = (; ) 2 H [0; 1) L : A + = : The coordinate projection of ) : The identi ed set for is given () onto H yields H(), the identi ed set for (Bertsimas and Tsitsiklis (1997, Chap.2) for an algorithm for projecting a polyhedron). 3.3 Unrevisable Prior Knowledge and a Class of Priors Let be a prior of and be the marginal probability measure on the su cient parameter space (; B) induced by and g(): (B) = ( (B)) for all B 2 B. Let x 2 X be sampled data. The posterior distribution of, denoted by F jx (), is obtained as F jx (A) = j (Aj)dF jx (); A 2 A, (3.4) where j (Aj) denotes the conditional distribution of given, and F jx () is the posterior distribution of. The posterior distribution of given in (3.4) shows that the prior distribution for marginalized to can be updated by data, while the conditional prior of given is never be updated by the data because the likelihood is at on () for any realizations of the sample. In this sense, the prior information marginalized to the su cient parameter can be interpreted as the revisable prior knowledge, and the conditional priors of given, j (j) : 2 can be interpreted as the unrevisable prior knowledge. If one wants to 13

14 summarize the posterior uncertainty of in the form of a probability distribution on (; A), as recommended in the Bayesian paradigm, he needs to have a single prior distribution of, which necessarily induces unique unrevisable prior knowledge j. If he could justify his choice of j by any credible prior information, the standard Bayesian updating (3.4) would yield a valid posterior distribution of : A challenging situation would arise if one is short of a credible prior distribution of. In this case, the researcher, who is aware that j will never be updated by data, might feel anxious in implementing the Bayesian inference procedure, because an uncon dently speci ed j can have a signi cant in uence to the subsequent posterior inference. The robust Bayes analysis in this paper speci cally focuses on such a situation, and introduce ambiguity for the conditional prior j (j) : 2 in the form of multiple priors. Speci cally, given a prior on (; B) speci ed by the researcher, consider the class of prior distributions of de ned by: M( ) = : ( (B)) = (B) for every B 2 B : M( ) consists of prior distributions of whose marginal distribution for the su cient parameters coincides with the prespeci ed. 5 This paper proposes to use M( ) as a prior input for the posterior analysis, meaning that, with accepting to specify a single prior distribution for the su cient parameters, I leave the conditional priors j unspeci ed and allow for arbitrary ones as long as () = R j(j)d yields a probability measure on (; A). 6 given. In the subsequent analysis, I shall not discuss how to select, and shall treat as The in uence of on the posterior of will diminish as the sample size increases, so the sensitivity issue of the posterior of is expected to be less severe when the sample size is moderate or large. 5 I thank Jean-Pierre Florens for suggesting this representation of the prior class. 6 Su cient parameters are de ned by examining the entire model fp (xj) : x 2 X ; 2 g, so that the prior class M( ) is, by construction, model dependent. This distinguishes the current approach from the standard robust Bayes analysis where a prior class represents the researcher s subjective assessment of his imprecise prior knowledge (Berger (1985)). 14

15 3.4 Posterior Lower and Upper Probabilities The Bayes rule is applied to each prior in M( ) to generate the class of posterior distributions of. Consider summarizing the posterior class by the posterior lower probability F jx () : A! [0; 1] and the posterior upper probability FjX () : A! [0; 1], de ned as F jx (A) inf F jx (A); 2M( ) FjX(A) sup F jx (A): 2M( ) Note that the posterior lower probability and the upper probability have a conjugate property, F jx (A) = 1 form. F jx (Ac ), so it su ces to focus on one of them in deriving their analytical In order to obtain F jx (), the following regularity conditions are assumed. Condition 3.1 (i) A prior for,, is proper and absolutely continuous with respect to a - nite measure on (; B). (ii) g : (; A)! (; B) is measurable and its inverse image () is a closed set in, -almost every 2. (iii) h : (; A)! (H; D) is measurable and H () = h ( ()) is a closed set in H, -almost every 2 : These conditions are imposed for () and H () to be interpreted as random closed sets induced by a probability measure on (; B). 7 The closedness of () and H () are implied, for instance, by continuity of g () and h (). Theorem 3.1 Assume Condition The inference procedure proposed in this paper can be implemented as long as the posterior of is proper. However, how to accommodate an improper prior for in the development of the analytical results is beyond the scope of this paper. 15

16 (i) For each A 2 A, F jx (A) = F jx (f : () Ag); (3.5) F jx(a) = F jx (f : () \ A 6= ;g) ; (3.6) where F jx (B); B 2 B, is the posterior probability measure of. (ii) De ne the posterior lower and upper probabilities of = h () by It holds F jx (D) inf F jx (h 1 (D)); 2M( ) FjX(D) sup F jx (h 1 (D)); for D 2 D. 2M( ) F jx (D) = F jx (f : H() Dg); F jx(d) = F jx (f : H() \ D 6= ;g): Proof. For a proof of (i), see Appendix A. For a proof of (ii), see equation (3.7). The expression for F jx (A) implies that the posterior lower probability on A calculates the probability that the set () is contained in subset A in terms of the posterior probability law of. On the other hand, the upper probability is interpreted as the posterior probability that the set () hits subset A. The second statement of the theorem provides a procedure for marginalizing the lower and upper probabilities of into those of the parameter of interest. The expressions of F jx (D) and FjX (D) are simple and easy to interpret: the lower and upper probabilities of = h() are the containment and hitting probabilities of the random sets obtained by projecting () through h(). This marginalization rule of the lower probability follows from F jx (D) = F jx (h 1 (D)) = F jx ( : () h 1 (D) ) = F jx (f : H() Dg). (3.7) 16

17 Note that, In the standard Bayesian inference, marginalization of the posterior of to is conducted by integrating the posterior probability measure of for, while in the lower probability inference, marginalization for corresponds to projecting random sets () via = h (). This stark contrast between the standard Bayes and the multiple prior robust Bayes inference highlights how the introduction of ambiguity changes the way of eliminating the nuisance parameters in the posterior inference. As is known in the literature (e.g., Huber (1973)), the lower probability of a set of probability measures is a monotone nonadditive measure (capacity). Furthermore, in the current speci cation of the prior class, the representation of the lower probability obtained in Theorem 3.1 implies that the resulting posterior lower and upper probabilities are supermodular and submodular, respectively. Corollary 3.1 Assume Condition 3.1. The posterior lower and upper probabilities of are supermodular and submodular, respectively. For A 1, A 2 2 A subsets in, F jx (A 1 [ A 2 ) + F jx (A 1 \ A 2 ) F jx (A 1 ) + F jx (A 2 ); FjX(A 1 [ A 2 ) + FjX(A 1 \ A 2 ) FjX(A 1 ) + FjX(A 2 ): Also, the posterior lower and upper probabilities of are supermodular and submodular, respectively. For D 1, D 2 2 D subsets in H, F jx (D 1 [ D 2 ) + F jx (D 1 \ D 2 ) F jx (D 1 ) + F jx (D 2 ); FjX(D 1 [ D 2 ) + FjX(D 1 \ D 2 ) FjX(D 1 ) + FjX(D 2 ): The results of Theorem 3.1 (i) can be seen as a special case of Wasserman s (1990) general construction of the posterior lower and upper probabilities. Whereas, one notable di erence from Wasserman s analysis is that, with prior class M, the lower probability of the posterior class becomes an 1-order monotone capacity (a containment functional of random 17

18 sets). 8 This plays a crucial role in simplifying the gamma minimax analysis considered in the next section. 4 Posterior Gamma-minimax Analysis for = h() In the standard Bayesian posterior analysis, a statistical decision problem involving (e.g., point estimation for ) is straightforward, minimizing the posterior risk. When the posterior information of is summarized by the class of posteriors, how should the optimal statistical action be solved? This section studies this problem by adapting the posterior gammaminimax analysis. Let a 2 H a be an action, where H a is an action space. In the case of the point estimation problem for, an action is interpreted as reporting a non-randomized point estimate for, where action space H a is a subset of H. Given an action a to be taken, and being the true state of nature, a loss function L(; a) : H H a! R + yields the cost to the decision maker of taking action a. I assume that the loss function L(; a) is non-negative. If a single prior for,, were given, the posterior risk would be de ned by ( ; a) L(; a)df jx (); (4.1) H where the rst argument in the posterior risk represents the dependence of the posterior of on the speci cation of the prior for. Our analysis involves multiple priors, so the class of posterior risks ( ; a) : 2 M( ) is considered. The posterior gamma-minimax criterion 9 ranks actions in terms of the worst case posterior risk (upper posterior risk): ( ; a) sup ( ; a), 2M( ) 8 Wasserman (1990, p.463) posed an open question asking which class of priors can assure the posterior lower probability to be a containment functional of random sets. Theorem 3.1 provides an answer to his open question in the situation where the model lacks identi ability. 9 In the robust Bayes literature, the class of prior distributions is often denoted by. This is why it is called the gamma-minimax criterion. upper probabilities, Unfortunately, in the literature of belief functions and lower and often denotes a set-valued mapping that generates the lower and upper probabilities. In this paper, we adopt the latter notational convention, but still refer to the decision criterion as the gamma-minimax criterion. 18

19 where the rst argument in the upper posterior risk represents the dependence of the prior class on a prior for. De nition 4.1 A posterior gamma-minimax action a x with respect to prior class M is an action that minimizes the upper posterior risk, i.e., ( ; a x) = inf a2h a ( ; a) = inf sup a2h a 2M( ) ( ; a). The gamma-minimax decision approach involves a favor for a conservative action that guards against the least favorable prior within the class, and it can be seen as a compromise of the Bayesian decision principle and the minimax decision principle. The next proposition shows that the upper posterior risk ( ; a) equals the Choquet expected loss with respect to the posterior upper probability. Proposition 4.1 Under Condition 3.1, the upper posterior risk satis es ( ; a) = L(; a)dfjx() = sup L(; a)df jx (), (4.2) 2H() whenever R L(; a)dfjx () < 1, where R L(; a)dfjx () is the Choquet integral. Proof. See Appendix A. The third expression in (4.2) shows that the posterior gamma-minimax criterion is written as the expectation of the worst-case loss function, sup 2H() L(; a), with respect to the posterior of. The supremum part stems from the ambiguity of : given, what the researcher knows about is only that it lies within the identi ed set H (), and, following the minimax principle, he forms the loss by supposing that the nature chooses the worst case in response to his/her action a. On the other hand, the expectation in represents the posterior uncertainty of the identi ed set H (): with the nite number of observations, the identi ed set of is known with some uncertainty as summarized by the posterior of. The posterior gamma-minimax criterion combines such ambiguity of with the posterior uncertainty of the identi ed set H () to yield a single objective function to be minimized The posterior gamma minimax action a x can be interpreted as a Bayes action for some posterior distributions in the class. For instance, in case of the quadratic loss, the saddle-point argument implies that the gamma-minimax action a x corresponds to the mean of a posterior distribution (Bayes action) that has maximal posterior variance in the class. 19

20 Although a closed-form expression of a x is not, in general, available, this proposition suggests a simple numerical algorithm for approximating a x using a random sample of from its posterior F jx. Let f s g S s=1 be S random draws of from posterior F jx. Then, a x can be approximated by ^a 1 x arg min a2h a S SX sup L(; a). 2H( s ) s=1 The gamma minimax decisions are usually dynamically inconsistent; a posteriori optimal gamma-minimax action does not coincide with an unconditional optimal gamma-minimax decision. This is also the case with out prior class, and this will imply that a x fails to be a Bayes decision with respect to any single prior in the class M. See Appendix B for an example and further discussion. As an alternative to the posterior gamma-minimax action, the gamma-minimax regret criterion may be considered (Berger (1985, p. 218), and Rios Insua, Ruggeri, and Vidakovic (1995)). Appendix B provides some analytical results of the posterior gamma-minimax regret analysis where the parameter of interest is a scalar and the loss function is quadratic, L(; a) = ( a) 2. There, it is shown that the posterior gamma-minimax regret decision can di er from the posterior gamma-minimax decision derived above, but that they converge to the same limit asymptotically. 5 Set Estimation of In the standard Bayesian inference, set estimation is often conducted by reporting the contour sets of the posterior probability density of (highest posterior density region). If the posterior information for is summarized by the lower and upper probabilities, how should we conduct set estimation of? 5.1 Posterior Lower Credible Region For 2 (0; 1), consider a subset C 1 H such that the posterior lower probability F jx (C 1 ) is greater than or equal to 1 : F jx (C 1 ) = F jx (H() C 1 )) 1. (5.1) 20

21 C 1 is interpreted as a set on which the posterior credibility of is at least 1, no matter which posterior is chosen within the class. If I drop the italicized part from this statement, I obtain the usual interpretation of the posterior credible region, so C 1 de ned in this way seems to be a natural extension of the Bayesian posterior credible regions to those of the posterior lower probability. multiple C 1 Analogous to the Bayesian posterior credible region, there are s that satisfy (5.1). For instance, given a posterior credibility region of with credibility 1, B 1, C 1 = [ 2B1 H () considered in the 2009 working paper version of Moon and Schorfheide (2011) satis es (5.1). In proposing set inference for, I resolve the multiplicity issue of set estimates by focusing on the smallest one, C 1 arg min Leb(C) (5.2) C2C s.t. F jx (H() C)) 1 ; where Leb(C) is the volume of subset C in terms of the Lebesgue measure and C is a family of subsets in H over which the volume-minimizing lower credible region is searched. I thereafter refer to C 1 de ned in this way as a posterior lower credible region with credibility 1. Note that focusing on the smallest set estimate has a decision theoretic justi cation; C 1 can be supported as a posterior gamma minimax action: 2 3 C 1 = arg min 4 sup L (; C) df jx 5 C2C 2M( ) with a loss function that penalizes the volume and non-coverage, L (; C) = Leb (C) + b () [1 1 C ()] ; where b () is a positive constant that depends on credibility level 1, and 1 C () is the indicator function for subset C. Here, the loss function is written in terms of parameter, so the object of interest is, rather than identi ed set H (). Finding C 1 is challenging if is multi-dimensional and no restriction is placed on class of subsets C. I therefore restrict our analysis to scalar and constrain C to the class of closed connected intervals. The next proposition shows how to obtain C 1. 21

22 Proposition 5.1 Let d : H D! R + measures distance from c 2 H to set H () in terms of d ( c ; H()) sup fk c kg. 2H() For each c 2 H, let r 1 ( c ) be the (1 )-th quantile of the distribution of d ( c ; H()) induced by the posterior distribution of, i.e., r 1 ( c ) inf r : F jx : d(c ; H()) r 1. Then, C 1 is a closed interval centered at c = arg min c 2H r 1 ( c ) with radius r 1 = r 1 ( c). Proof. See Appendix A. 5.2 Asymptotic Properties of the Posterior Lower Credible Region In this section, I examine the large-sample behavior of the posterior lower probability in an intervally identi ed case, where H () R is a closed bounded interval for almost all 2. From now on, I make the sample size explicit in the notation: a size n sample X n is generated from its sampling distribution P X n j 0, where 0 denotes the value of the su cient parameters that corresponds to the true data-generating process. The maximum likelihood estimator for is denoted by ^. Provided that the posterior of is consistent 11 to 0 and the set-valued map H ( 0 ) is continuous at = 0 in terms of the Hausdor metric d H, it can be shown that random sets H (), represented by the posterior lower probability F jx n (), converges to true identi ed set H ( 0 ) in the sense of lim n!1 F jx n (f : d H (H () ; H ( 0 )) > g) = 0 for almost 11 Posterior consistency of means that lim n!1 F jx n (G) = 1 for every G open neighborhood of 0 for almost every sampling sequence. For nite dimensional, this posterior consistency for is implied by a set of higher-level conditions for the likelihood of. We do not list up all those conditions here for the sake of brevity. See Section 7.4 of Schervish (1995) for details. 22

23 every sampling sequences of fx n g. Given such posterior consistency of, we analyze the asymptotic coverage property of the lower credible region C 1. The following set of regularity conditions are imposed to show the asymptotic correct coverage property of C 1. Condition 5.1 (i) The parameter of interest is a scalar and the identi ed set H() is -almost surely a non-empty and connected interval, H () = [ l (); u ()], the true identi ed set H ( 0 ) = [ l ( 0 ); u ( 0 )] is a bounded interval. 1 l () u () 1, and (ii) For sequence a n! 1, random variables ^L = a n l ( 0 ) l (^) and ^U = a n u ( 0 ) u (^) converges in distribution to bivariate random variables (L; U), whose cumulative distribution function J () on R 2 is Lipschitz continuous and monotonically increasing in the sense of J (c l ; c u ) < J (c l + ; c u + ) for any > 0. (iii) De ne random variables L n () = a n l () l (^) and U n () = a n u () whose distribution is induced by the posterior distribution of given sample X n. The cumulative distribution function of (L n () ; U n ()) given X n denoted by J n () is continuous almost surely under ^p (x n j 0 ) for all n: (iv) At each c (c l ; c u ) 2 R 2, the cumulative distribution function of (L n () ; U n ()) given X n, J n (c), converges in probability under P X n j 0 to J (c). Conditions 5.1 (ii) and (iv) imply that the estimators for the lower and upper bounds of H () attain the Bernstein von Mises property: the sampling distribution of the bound estimators and the posterior distribution of the bounds coincide asymptotically in the sense of Theorem in Schervish (1995). and (iv), with a n In case of nite dimensional, Condition 5.1 (ii) = p n and bivariate normal (L; U), are implied from the following set of assumptions: (a) the regularity of the likelihood of and the asymptotic normality of p n ^, (b) puts a positive probability on every open neighborhood of 0 and s density is smooth at 0, and (c) the applicability of the delta method to l () and u () at = 0 with non-zero rst derivatives See Schervish (1995, Section 7.4) for further detail on these assumptions. u (^), 23

24 The next proposition establishes the large-sample coverage property of the posterior lower credible region C 1. Theorem 5.1 (Asymptotic Coverage Property) Assume Conditions 3.1 and 5.1. C 1 can be interpreted as frequentist con dence intervals for the true identi ed set H ( 0 ) with a pointwise asymptotic coverage probability (1 lim P X n!1 n j 0 (H ( 0 ) C 1 ) = 1 : Proof. See Appendix A. C 1 ), This result shows that, for the interval-identi ed, the posterior lower credible region achieves the exact desired frequentist coverage for the identi ed set asymptotically (Horowitz and Manski (2000), Chernozhukov, Hong, and Tamer (2007), and Romano and Shaikh (2010)). It is worth noting that the posterior lower credible region C 1 di ers from the con dence intervals for the parameter of interest, as considered in Imbens and Manski (2004) and Stoye (2009); in case H ( 0 ) is an interval, C 1 will be asymptotically wider than the frequentist con dence interval for. This implies that the set of priors M is too large to interpret C 1 as the frequentist s con dence interval for. It is also worth noting that the asymptotic coverage probability presented in Theorem 5.1 is in the sense of pointwise asymptotic coverage rather than an asymptotic uniform coverage over 0. The frequentist literature has stressed the importance of the uniform coverage property of interval estimates in order to ensure that the intervals estimates can have an accurate coverage probability in a nite sample situation (Imbens and Manski (2004), Andrew and Guggenberger (2009), Stoye (2009), Romano and Shaikh (2010), Andrew and Soares (2010), among many others). Examining whether or not the posterior lower credible region constructed above can attain a uniformly valid coverage probability for the identi ed set is beyond the scope of this paper and is left for future research. We note that Condition 5.1 (iv) is a quite delicate condition as illistrated by the following counterexample. Example 5.1 Let the identi ed set be given by H () = [max f 1 ; 2 g ; min f 3 ; 4 g]. type of bound commonly appears in the intersection bound analysis (Manski (1990)), and has This 24

25 attracted considerable attention in the literature (Hirano and Porter (2011), Chernozhukov, Lee, and Rosen (2011)). Condition 5.1 (iv) does not hold in this class of models when the true values of the arguments in the minimum or maximum happen to be equal. Let us focus on the lower bound l () = max f 1 ; 2 g. Assume that the maximum likelihood estimators ^ 1 and ^ 2 are independent and ^ 1 N 10 ; n 1 and ^2 N 20 ; 1 n with 10 = 20. As for the posterior of 1 and 2, assume that 1 jx n N ^1 ; 1 and n 2 jx n N ^2 ; 1. In this case, the sampling distribution of L = p n n l ( 0 ) l (^) and the posterior distribution of L n () = p n l () l (^) are obtained as 8 < L max : = 8 < ; ; Ln () jx n min : 1 + ^ 2 + ^ where ( 1 ; 2 ) are independent standard normal variables, ^ = p n ^1 ^2, jaj = min f0; ag, and jaj + = max f0; ag. The posterior distribution of L n () fails to converge to the sampling distribution of p n l (^) l ( 0 ) due to the non-vanishing ^. Note that, even when ^ happens to be zero, L n () s posterior distribution di ers from the sampling distribution of L. C = ; ; will estimate H ( 0 ) with inward bias, and the coverage probability for H ( 0 ) will be lower than the nominal coverage. A lesson from this example is that, despite the explicit introduction of ambiguity in the form of prior class M and the decision theoretic justi cation behind the construction of C 1 procedure based on C 1, the robust Bayes posterior inference does not correct the frequentist bias issue in the intersection bounds analysis. In contrast, the correct coverage will be attained if C 1 = [ 2B1 H () is used as a robusti ed posterior credible region. 6 Concluding Remarks This paper proposes a framework of a robust Bayes analysis for set-identi ed models in econometrics. I demonstrate that the posterior lower probability obtained from the prior class M can be interpreted as the posterior probability law of the identi ed set (Theorem 3.1). This robust Bayesian way of generating and interpreting the identi ed set as an a posteriori random object has not been investigated in the literature, This highlights the 25

26 seamless links among partial identi cation analysis, robust Bayes inference, and random set theory, and o ers a uni ed framework of statistical decision and inference for set-identi ed parameters from the conditional perspective. I employ the posterior gamma-minimax criterion to formulate and solve for a statistical decision with multiple posteriors. The objective function of the gamma-minimax criterion integrates the ambiguity associated with the set identi cation and posterior uncertainty of the identi ed set into a single objective function. It leads to a numerically solvable posterior gamma-minimax action, as long as the identi ed sets H () can be simulated from the posterior of. The posterior lower probability is a non-additive measure, so one complication of the lower probability inference is that we cannot plot it as we would do for the posterior probability densities. To visualize it and conduct a set estimation in a decision-theoretically justi able way, I propose the posterior lower credible region. For an interval-identi ed parameter, I derive the conditions that the posterior lower credible region with credibility (1 ) can be interpreted as an asymptotically valid frequentist con dence interval for the identi ed set with coverage (1 ). This claim can be seen as an extension of the celebrated Bernsteinvon Mises theorem to the multiple prior Bayesian inference via the lower probability, and exempli es a situation where the robust Bayesians can accomplish a compromise of the Bayesian and frequentist inference. Appendix A Lemmas and Proofs In this appendix, I rst demonstrate that the set-valued mappings () and H () de ned in the main text are closed random sets (measurable and closed set-valued mappings) induced by a probability measure on (; B). Lemma A.1 Assume (; A) and (; B) are complete separable metric spaces. Under Condition 3.1, () and H () are random closed sets induced by a probability measure on (; B), 26